Fuzzy Criteria and Fuzzy Rules in Subjective Evaluation A General Discussion

This position paper emphasizes the interest of fuzzy set-based methods in subjective evaluation problems, where complex situations, or objects, have to be classified into categories, or have to be assessed under the form of global estimates. More particularly, different types of approaches to these problems, such as multiple-criteria aggregation, rule-based inference, case-based reasoning, are reviewed and put in perspective. Important issues in subjective evaluation, such as the expressibility of an approach with respect to the description of the evaluation process, are pointed out. The paper illustrates this general overview by means of some examples taken from the literature. 1 SUBJECTIVE EVALUATION A subjective evaluation problem aims at providing a global assessment of a (complex) situation or object. The situation or the object under consideration is supposed to be described in terms of various attributes, and the assessment may be made either in a discrete way by classifying the situation according to given categories, or in an absolute way by computing a global rate, taking its value in a continuous scale. It may also correspond to the extrapolation of the value of some parameter attached to the situation. A daylife example of subjective evaluation problem is the pricing of a house from the knowledge of its surface, of its location w.r.t. downtown, its age, and some other attributes. This problem generally amounts to the computation of an estimated price, but viewing it as a discrete evaluation problem, it would mean classifying the price of the house with respect to context-dependent categories like "cheap", "moderate", "expensive", etc. Subjective evaluation problems can be encountered in a great number of areas. Let us, for instance, mention the evaluation of complex notions such as comfort (Levrat et al., 1997), creditworthiness of a customer in a bank (Zimmermann and Zysno, 1983), the assessment of damages, of defects in quality control (Ménage, 1996), the design of attractive products (e.g., Grabisch et al., 1997, in cosmetics), the selection of candidates from their profiles, the evaluation of recommended amounts of calories in dietetics in specific situations,... Fuzzy set methods have been advocated for a long time in subjective evaluation for representing the result of the evaluation (by means of fuzzy estimates, or fuzzy classes), and/or for describing the evaluation process itself (even if all the attributes describing the situation or the object are precisely known). A general introduction to fuzzy set methods in subjective evaluation can be found in the recent book (Club CRIN Logique Floue, 1997). More generally, subjective evaluation problems have an important place in Fuzzy Information Engineering (Dubois et al., 1997) in relation with decision issues. Formally speaking, the subjective evaluation problem can be viewed as the synthesis, the identification of a function which maps the attribute values describing the situation to evaluate into a discrete domain (classification), or a continuous one (absolute evaluation). More generally, we may look for the degree of membership of the situation to a category, or have a function yielding a fuzzy evaluation. This function is in general not available as such, but is implicitly, and partially, described in terms of criteria, or by means of expert rules, or through some fuzzy algorithm. It may also happen that the function is only partially known by exemplification through prototypical examples of situations for which the evaluation is available. We might also think of a neural net approach for synthesizing the function from a set of training examples, but then, we would not take advantage of the available knowledge on the evaluation process if any, and there will be no basis for explaining the evaluation results to a user. In the following, we briefly discuss the different knowledge-based approaches that have been just mentioned and the role of fuzzy set methods in each of them, before concluding on the interest of a unified view of the different approaches. 2 MULTICRITERIA VIEW The evaluation function, say f, which relates the attribute values x 1, ..., xn (assumed to be precisely known for simplicity) describing the situation or the object under consideration to its estimation y, is supposed in the multiplecriteria view, to be obtained by an appropriate aggregation of the evaluations of the attribute values x i by means of criteria Ci, for i = 1,n. Namely, f can be decomposed into local evaluation functions of each x i : f(x1, ..., xn) = φ(C1(x1), ..., Cn(xn)). In the above expression, φ is not necessarily an associative aggregation operation. Different specific aggregation operations may be used for combining evaluations over subsets of the criteria (triangular-norms, averages, etc.), or φ may involve even more sophisticated functions taking into account some interaction between the criteria. Such an approach raises several questions about – the choice of proper scales (what kind? qualitative or numerical scale?), their commensurability, and the meaningfulness of the aggregation operations w.r.t. the scale; – the practical elicitation of the membership functions, and of the appropriate operations (compensatory, or purely logical conjunctions, for instance); see (Dubois and Prade, 1988) on this latter point, where the elicitation of aggregation operations is based on the knowledge of the decision's maker's behavior in well-contrasted situations; – the modelling of the importance (by means of weights or thresholds) of the criteria, and more generally of the interaction between criteria (Carlsson and Fullér, 1994; Grabisch, 1997). A pioneering example of such an approach is the assessment of the credit-worthiness of a bank customer (Zimmermann and Zysno, 1983; see also Zimmermann, 1997). This approach has been used in many other applications, including financial analysis. Another worth mentioning example of this type of approach to a completely different application, is the cutting of long woods in vine pruning (Tisseyre et al., 1996, 1997). In such a problem, the professional pruner for deciding what long wood to keep for next year, takes into account six main criteria (reflecting experts' preferences), the diameter, the length, the direction, the linearity of the cane and its X and Y positions with respect to the axis of the vine trunk and the wire respectively. Tisseyre et al. (1996) have succeeded in identifying an aggregation function for combining all these criteria and obtaining evaluation results which are well in agreement with expert opinions, in standard situations. In special cases where one or several of the important criteria are not satisfied at all, the way the expert selects the long wood which will not be cut, largely differs from his approach to standard situations. Tisseyre et al. (1996, 1997) use a fuzzy rule-based approach for describing the long wood selection in such special cases. It raises the issue of integrating a multiple-criteria view and a rule-based approach in the same framework. Conditional prioritized requirements are a particular instance of rule-based expressions which can be captured in the framework of a multiple-criteria aggregation approach. This type of problem has been encountered in database querying systems by Lacroix and Lavency (1987) who deal with requirements of the form "P 1 should be satisfied, and among the solutions to P1 (if any) the ones satisfying P 2 are preferred, and among those satisfying both P 1 and P2, those satisfying P3 are preferred, and so on", where P 1, 2, P3..., are binary constraints for simplicity. It should be understood in the following way: satisfying P2 if P1 is not satisfied is of no interest; satisfying P 3 if P2 is not satisfied is of no use even if P1 is satisfied. Thus, there is a hierarchy between the requirements. A request looking for candidates such that "if they are not graduated they should have professional experience, and if they have professional experience, they should preferably have communication abilities", is an example where only conditional constraints, organized in a hierarchical way, take place. It can be represented by an expression of the form min[max(Prof.exp.(d), Grad.(d)), max(Com.ab.(d), 1 – min(1 – Grad.(d), Prof.exp.(d), α)] so that if d has professional experience and communication abilities, d completely satisfies the request, as well as if d is graduated; d satisfies the request to the degree 1 – α if d is not graduated and has professional experience only. d does not satisfy the request at all if d is neither graduated nor has professional experience (even if d has communication abilities). See (Dubois and Prade, 1996b) for details on this approach, where a conditional requirement of the form "if P is true then d should be C" is represented (and estimated) by max(C(d), 1 – P(d)) where P(d) = 1 if the choice d satisfies the context P and is 0 otherwise, and where C(d) rates d according to criterion C (outside context P, C(d) is not taken into account).